Question

given: \\( \overline{ab} \cong \overline{bc} \\); \\( \overline{bm} \\) is an altitude of \\( \triangle abc \\). prove: point \\( m \\) is the midpoint of \\( \overline{ac} \\). 1. \\( \overline{bm} \\) is an altitude of \\( \triangle abc \\). 1. given 2. \\( \overline{bm} \perp \overline{ac} \\) 2. definition of an altitude 3. \\( \angle amb \\) and \\( \angle cmb \\) are right angles. 3. click here to insert 4. click here to insert 4. definition of a right triangle

Explanation:

Step1: Determine the reason for statement 3

Since \(\overline{BM}\perp\overline{AC}\) (from statement 2), by the definition of perpendicular lines (if two lines are perpendicular, then the angles formed are right angles), \(\angle AMB\) and \(\angle CMB\) are right angles. So the reason for statement 3 is "Definition of perpendicular lines".

Step2: Determine the statement for step 4

Since \(\angle AMB\) and \(\angle CMB\) are right angles (from statement 3), and by the definition of a right - triangle (a triangle with one right angle is a right - triangle), \(\triangle AMB\) and \(\triangle CMB\) are right triangles. So the statement for step 4 is "\(\triangle AMB\) and \(\triangle CMB\) are right triangles".

Answer:

1. Definition of perpendicular lines; 4. \(\triangle AMB\) and \(\triangle CMB\) are right triangles.